en>PigFlu Oink: fmt headline levels (to start with "==", WikiCheck check #7

2009-05-23T12:04:39Z

fmt headline levels (to start with "==", WikiCheck check #7

New page

[[Image:DigitalSundialPatentIllustration.png|thumb|right|A patent illustration of a fractal digital sundial.]]

A '''digital sundial''' is a clock that indicates the current time with numerals formed by the sunlight striking it. Like a classical [[sundial]], the device is purely passive and contains no moving parts. It uses no electricity nor other artificial sources of energy.
The digital display changes as the sun advances in its daily course.

==Technique==
There are two basic types of digital sundials. One construction uses [[Waveguide (optics)|optical waveguides]], while the other is inspired by [[fractal]] geometry.

===Optical fiber sundial===

Sunlight enters into the device through a slit and moves as the sun advances. The sun's rays shine on ten linearly distributed sockets of optical waveguides that transport the light to a [[seven-segment display]]. Each socket fiber is connected to a few segments forming the digit corresponding to the position of the sun.<ref>[http://www.hineslab.com/DigitalSundial.html US patent belongs to HinesLab Inc. (USA)]</ref>

===Fractal sundial===
[[File:Digital sundial.jpg|thumb|This sundial by ''Digital sundials international'' uses just two masks and a [[plexiglass]] layer.]]

The theoretical basis for the other construction comes from fractal geometry.<ref>{{cite book
| last = Falconer
| first = Kenneth
| title = Fractal Geometry: Mathematical Foundations and Applications
| publisher = John Wiley & Sons, Ltd.
| year = 2003
| isbn = 0-470-84862-6
| nopp = true
| page = xxv}}</ref> For the sake of simplicity we describe a two-dimensional (planar) version.
Let <math> L_\theta</math> denote a straight line passing through the origin of a [[Cartesian coordinate system]] and making angle <math> \theta\in [0,\pi)</math> with the x-axis.
For any <math> F\subset \mathbb{R}^2</math> define <math>\mathrm{proj}_\theta F</math> to be the perpendicular projection of ''F'' on the line <math>L_\theta</math>.

====Theorem====

Let <math>G_\theta \subset L_\theta</math>, <math>\theta\in[0,\pi)</math> be a family of any sets such that <math>\bigcup_\theta G_\theta</math> is a [[measurable set]] in the plane.
Then there exists a set <math>F\subset \mathbb{R}^2</math>
such that
* <math> G_\theta \subset \mathrm{proj}_\theta F </math>;
* the measure of the set <math>(\mathrm{proj}_\theta F)\setminus
G_\theta</math> is zero for almost all <math>\theta\in[0,\pi).</math>
There exists a set with prescribed projections in ''almost'' all directions. This theorem can be generalized to three-dimensional space. For a non-trivial choice of the family <math> G_\theta</math>, the set ''F'' described above is a fractal.

====Application====

Theoretically, it is possible to build a set of masks that produce shadows in the form of digits, such that the display changes as the sun moves. This is the fractal sundial.

The theorem was proved in 1987 by [[Kenneth Falconer (mathematician)|Kenneth Falconer]]. Four years later it was described in ''[[Scientific American]]'' by [[Ian Stewart (mathematician)|Ian Stewart]].<ref>Ian Stewart, [[Scientific American]], 1991, pages 104-106, ''Mathematical Recreations'' -- ''What in heaven is a digital sundial?''.</ref> The first prototype of the device was constructed in 1994. In 1998 for the first time a fractal sundial was installed in a public place ([[Genk%2C_Belgium#Sights|Genk, Belgium]]).<ref>[http://www.fransmaes.nl/genk/welcome-e.htm Sundial park in Genk, Belgium]</ref> There exist window and tabletop versions as well.<ref>[http://www.digitalsundial.com/patent.html US and German patent belongs to Digital Sundials International]</ref>

==References==
{{Reflist}}

[[Category:Sundials]]

Calculated Ignition Index - Revision history

en>PigFlu Oink: fmt headline levels (to start with "==", WikiCheck check #7